Galois action on the homology of Fermat curves

Abstract: In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as a Galois module for the action of the absolute Galois group $G_{\mathbb{Q}(\zeta_n)}$. In particular, when $n$ is an odd prime $p$, he shows that the action of $G_{\mathbb{Q}(\zeta_p)}$ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial $1-(1-x^p)^p$. If $p$ satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over $\mathbb{Q}(\zeta_p)$ is an elementary abelian $p$-group of rank $(p+1)/2$. Using an explicit basis for this Galois group, we completely compute the relative homology, the homology, and the homology of an open subset of the degree $3$ Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.